Executive Summary

Stock: Red king crab (RKC), Paralithodes camtschaticus, in Bristol Bay, Alaska.
Catches: The domestic RKC fishery began to expand in the late 1960s and peaked in 1980 with a catch of 129.95 million lbs (58,943 t). The catch declined dramatically in the early 1980s and remained at low levels during the last three decades. Catches during recent years until 2010/11 were among the high catches in last 15 years. The retained catch in 2015/16 was about 10 million lbs (4,500 t), similar to the catch in 2014/15. The magnitude of bycatch from groundfish trawl fisheries has been stable and small relative to stock abundance during the last 10 years.
Stock biomass: Estimated mature biomass increased dramatically in the mid 1970s and decreased precipitously in the early 1980s. Estimated mature crab abundance had increased during 1985-2009 with mature females being about three times more abundant in 2009 than in 1985 and mature males being about two times more abundant in 2009 than in 1985. Estimated mature abundance has steadily declined since 2009.
Recruitment: Estimated recruitment was high during 1970s and early 1980s and has generally been low since 1985 (1979 year class). During 1984-2016, only in 1984, 1986, 1995, 1999, 2002 and 2005 were estimated recruitments above the historical average for 1976-2016. Estimated recruitment was extremely low during the last 10 years.
Management performance: Status and catch specifications (1,000 t) (scenario 2) are given below. In recent assessments, estimated total male catch has been determined as the sum of fishery-reported retained catch, estimated male discard mortality in the directed fishery, and estimated male bycatch mortality in the groundfish fisheries, as these have been the only sources of non-negligible fishing mortality to consider. The stock was above the minimum stock-size threshold (MSST) in 2016/17 and is hence not overfished. Overfishing did not occur in 2016/17 (Tables and ).

Basis for the OFL: Estimated mature-male biomass (MMB) on 15 February is used as the measure of biomass for this Tier 4 stock, with males measuring 105 mm CL or more considered mature. The $B_\mathit{MSY}$ proxy is obtained by averaging estimated MMB over a specific reference time period, and current CPT/SSC guidance recommends using the full assessment time frame as the default reference period (Table ).

A. Summary of Major Changes

Changes in Management of the Fishery

There were no new changes in management of the fishery.

Changes to the Input Data

The new 2016 NMFS trawl survey data and BSFRF side-by-side trawl survey data during 2013-2016 were used.
Catch and biomass data were updated to include the 2016/17 information.

Changes in Assessment Methodology

This assessment was done using Gmacs. There are several differences between the Gmacs assessment model and the previous model. One of the major differences being that natural and fishing mortality are continuous within r M[[2]]$nseason discrete seasons. Season length in Gmacs is controlled by changing the proportion of natural mortality that is applied during each season. A detailed outline of the Gmacs implementation of the BBRKC model is provided in Appendix A.

Changes in Assessment Results

B. Responses to SSC and CPT Comments

CPT and SSC Comments on Assessments in General

Comment: To come

Response:

CPT and SSC Comments Specific to the BBRKC Stock Assessment

Comment: * [to come ] The SSC and CPT requested the following models for review at the spring 2016 meeting:*

Base: try to match 2015 model but prevent dome shaped selectivity
Base + add CV for both surveys
Above + Francis re-weighting
Above + remove M spike

Response: [to come ] Models 1, 3, and 4 are all included and evaluated in this document as the Gmacs base, Gmacs Francis, and Gmacs M scenarios. Model 2 was not included in this document for two reasons. Firstly, if doing Francis iterative re-weighting then additional CV should not be added as well (as the two methods basically do the same thing). Secondly, the SSC recommended against the model runs with additional CV (see the comment from the SSC below).

Comment: * [to come ] The SSC is not convinced that the model runs with extra CV are very informative. The inclusion of extra CV seems to be rather arbitrary based on the numbers of points that fall within confidence intervals estimated from trawl surveys. The SSC recommends coming up with some alternative way to consider extra variability, which could be informed by simulation testing.*

Response: [to come ] All model runs that estimate additional CV were dropped from this document. Instead we provide two model runs that use the Francis iterative re-weighting method to re-weight the length-frequency data relative to the abundance indices. These runs are the Gmacs Francis, and Gmacs force scenarios. The final Gmacs scenario (Gmacs force) is an exploratory model run that upweights both the trawl-survey and pot survey abundance indices (it upweights the pot survey more than the trawl survey).

C. Introduction

Scientific Name

Red king crab (RKC), Paralithodes camtschaticus, in Bristol Bay, Alaska.

Distribution

Red king crab inhabit intertidal waters to depths >200 m of the North Pacific Ocean from British Columbia, Canada, to the Bering Sea, and south to Hokkaido, Japan, and are found in several areas of the Aleutian Islands, eastern Bering Sea, and the Gulf of Alaska.

Stock Structure

The State of Alaska divides the Aleutian Islands and eastern Bering Sea into three management registration areas to manage RKC fisheries: Aleutian Islands, Bristol Bay, and Bering Sea (Alaska Department of Fish and Game (ADF&G) 2012). The Bristol Bay area includes all waters north of the latitude of Cape Sarichef (54°36’ N lat.), east of 168°00’ W long., and south of the latitude of Cape Newenham (58°39’ N lat.) and the fishery for RKC in this area is managed separately from fisheries for RKC outside of this area; i.e., the red king crab in the Bristol Bay area are assumed to be a separate stock from red king crab outside of this area. This report summarizes the stock assessment results for the Bristol Bay RKC stock.

Life History

Red king crab have a complex life history. Fecundity is a function of female size, ranging from several tens of thousands to a few hundreds of thousands (Haynes 1968; Swiney et al. 2012). The eggs are extruded by females, fertilized in the spring, and held by females for about 11 months (Powell and Nickerson 1965). Fertilized eggs are hatched in the spring, most during April-June (Weber 1967). Primiparous females are bred a few weeks earlier in the season than multiparous females. Larval duration and juvenile crab growth depend on temperature (Stevens 1990; Stevens and Swiney 2007). Male and female RKC mature at 5–12 years old, depending on stock and temperature (Loher et al. 2001; Stevens 1990) and may live >20 years (Matsuura and Takeshita 1990). Males and females attain a maximum size of 227 and 195 mm carapace length (CL), respectively (Powell and Nickerson 1965). Female maturity is evaluated by the size at which females are observed to carry egg clutches. Male maturity can be defined by multiple criteria including spermataphore production and size, chelae vs. carapace allometry, and participation in mating in situ (reviewed by Webb 2014). For management purposes, females >89 mm CL and males >119 mm CL are assumed to be mature for Bristol Bay RKC. Juvenile RKC molt multiple times per year until age 3 or 4; thereafter, molting continues annually in females for life and in males until maturity. Male molting frequency declines after attaining functional maturity.

Fishery

The RKC stock in Bristol Bay, Alaska, supports one of the most valuable fisheries in the United States. A review of the history of the Bristol Bay RKC fishery is provided in Fitch et al. (2012) and Otto (1989). The Japanese fleet started the fishery in the early 1930s, stopped fishing from 1940 to 1952, and resumed the fishery from 1953 until 1974. The Russian fleet fished for RKC from 1959 to 1971. The Japanese fleet employed primarily tanglenets with a very small proportion of catch from trawls and pots. The Russian fleet used only tanglenets. United States trawlers started fishing Bristol Bay RKC in 1947, but the effort and catch declined in the 1950s. The domestic RKC fishery began to expand in the late 1960s and peaked in 1980 with a catch of 129.95 million lbs (58,943 t), worth an estimated $115.3 million ex-vessel value. The catch declined dramatically in the early 1980s and has remained at low levels during the last two decades (Table 1). After the early 1980s stock collapse, the Bristol Bay RKC fishery took place during a short period in the fall (usually lasting about a week) with the catch quota based on the stock assessment conducted the previous summer (Zheng and Kruse 2002). Beginning with the 2005/2006 season, new regulations associated with fishery rationalization resulted in an increase in the duration of the fishing season (October 15 to January 15). With the implementation of crab rationalization, historical guideline harvest levels (GHL) were changed to a total allowable catch (TAC). Before rationalization, the implementation errors were quite high for some years and total actual catch from 1980 to 2007 was about 6% less than the sum of GHL/TAC over that period.

Fisheries Management

King and Tanner crab stocks in the Bering Sea and Aleutian Islands are managed by the State of Alaska through a federal king and Tanner crab fishery management plan (FMP). Under the FMP, management measures are divided into three categories: (1) fixed in the FMP, (2) frame worked in the FMP, and (3) discretion of the State of Alaska. The State of Alaska is responsible for determining and establishing the GHL/TAC under the framework in the FMP. Harvest strategies for the Bristol Bay RKC fishery have changed over time. Two major management objectives for the fishery are to maintain a healthy stock that ensures reproductive viability and to provide for sustained levels of harvest over the long term (ADF&G 2012). In attempting to meet these objectives, the GHL/TAC is coupled with size-sex-season restrictions. Only males ≥6.5-in carapace width (equivalent to 135-mm carapace length, CL) may be harvested and no fishing is allowed during molting and mating periods (ADF&G 2012). Specification of TAC is based on a harvest rate strategy. Before 1990, harvest rates on legal males were based on population size, abundance of prerecruits to the fishery, and postrecruit abundance, and rates varied from less than 20% to 60% (Schmidt and Pengilly 1990). In 1990, the harvest strategy was modified, and a 20% mature male harvest rate was applied to the abundance of mature-sized (≥120-mm CL) males with a maximum 60% harvest rate cap of legal (≥135-mm CL) males (Pengilly and Schmidt 1995). In addition, a minimum threshold of 8.4 million mature-sized females (≥90-mm CL) was added to existing management measures to avoid recruitment overfishing (Pengilly and Schmidt 1995). Based on a new assessment model and research findings (Zheng et al. 1995a, 1995b, 1997a, 1997b), the Alaska Board of Fisheries adopted a new harvest strategy in 1996. That strategy had two mature male harvest rates: 10% when effective spawning biomass (ESB) is between 14.5 and 55.0 million lbs and 15% when ESB is at or above 55.0 million lbs (Zheng et al. 1996). The maximum harvest rate cap of legal males was changed from 60% to 50%. A threshold of 14.5 million lbs of ESB was also added. In 1997, a minimum threshold of 4.0 million lbs was established as the minimum GHL for opening the fishery and maintaining fishery manageability when the stock abundance is low. The Board modified the current harvest strategy by adding a mature harvest rate of 12.5% when the ESB is between 34.75 and 55.0 million lbs in 2003 and eliminated the minimum GHL threshold in 2012. The current harvest strategy is illustrated in Figure 1.

D. Data

Summary of New Information

Data used in this assessment have been updated to include the most recently available fishery and survey numbers. This assessment makes use of two new survey data points including the 2016 NMFS trawl-survey estimate of abundance, and the 2016 ADF&G pot survey CPUE. Both of these surveys have associated size compositon data. The assessment also uses updated 1993-2015 groundfish and fixed gear bycatch estimates based on AKRO data. The 2015/16 directed fishery catch data and associated size composition data were also used. The data used in each of the new models is shown in Figure .

$Data extent for the BBRKC assessment.\label{fig:data_extent}$

Data extent for the BBRKC assessment.

Major Data Sources

Major data sources used in this assessment include annual directed-fishery retained-catch statistics from fish tickets (1978/79-1998/99, 2009/10-2012/13, and 2014/15-2015/16; Table ); results from the annual NMFS eastern Bering Sea trawl survey (1978-2016; Table ); results from the triennial ADF&G bbrkC pot survey (every third year during 1995-2013), the 2015 pot survey, and the 2016 pot survey (Table ); size-frequency information from ADF&G crab-observer pot-lift sampling (1990/91-1998/99, 2009/10-2012/13, and 2014/15-2015/16; Table ); and NMFS groundfish-observer bycatch biomass estimates (1992/93-2015/16; Table ).

XXXFigure maps stations from which SMBKC trawl-survey and pot- survey data were obtained. Further information concerning the NMFS trawl survey as it relates to commercial crab species is available in Daly et al. (2014); see Gish et al. (2012) for a description of ADF&G SMBKC pot-survey methods. It should be noted that the two surveys cover different geographic regions and that each has in some years encountered proportionally large numbers of male blue king crab in areas where the other is not represented (Figure ). Crab-observer sampling protocols are detailed in the crab-observer training manual (ADF&G 2013). Groundfish SMBKC bycatch data come from NMFS Bering Sea reporting areas 521 and 524 (Figure ). Note that for this assessment the newly available NMFS groundfish observer data reported by ADF&G statistical area was not used.

Other Data Sources

XXXRecent model configurations developed for SMBKC makes use of a growth transition matrix based on Otto and Cummiskey (1990), the same growth transition matrix is used in this assessment. Other relevant data sources, including assumed population and fishery parameters, are presented in Appendix A, which also provides a detailed description of the model configuration used for this assessment.

Excluded Data Sources

Groundfish bycatch size-frequency data are available for selected years. These data were used in model-based assessments prior to 2011. However, they have since been excluded because these data tend to be severely limited: for example, 2012/13 data include a total of just 4 90 mm+ CL male blue king crab from reporting areas 521 and 524.

E. Analytic Approach

History of Modeling Approaches for this Stock

XXXA four-stage catch-survey-analysis (CSA) assessment model was used before 2011 to estimate abundance and biomass and prescribe fishery quotas for the SMBKC stock (2010 SAFE; Zheng et al. 1997). The four-stage CSA is similar to a full length-based analysis, the major difference being coarser length groups, which are more suited to a small stock with consistently low survey catches. In this approach, the abundance of male crab with a CL of 90 mm or above is modeled in terms of four crab stages: stage 1: 90-104 mm CL; stage 2: 105-119 mm CL; stage 3: newshell 120-133 mm CL; and stage 4: oldshell $\ge$ 120 mm CL and newshell $\ge$ 134 mm CL. Motivation for these stage definitions comes from the fact that for management of the SMBKC stock, male crab measuring at least 105 mm CL are considered mature, whereas 120 mm CL is considered a proxy for the legal size of 5.5 in carapace width, including spines. Additional motivation for these stage definitions comes from an estimated average growth increment of about 14 mm per molt for SMBKC (Otto and Cummiskey 1990).

Assessment Methodology

The 2016 SMBKC assessment model makes use of the modeling framework Gmacs. The aim when developing this model was to first provide a fit to the data that best matched the 2015 SMBKC stock assessment model. A detailed description of the Gmacs model and its implementation is presented in Appendix A.

Model Selection and Evaluation

Five different Gmacs model scenarios were considered, in this document results from these models and the 2015 model are compared. The models inlcude:

2015 Model: the 2015 approach with a correction¹. This modification was made prior to comparisons (note that this modification caused the NMFS trawl survey selectivity to exceed 1 for stage-2 crab).
Gmacs match: tries to match as closely as possible with the 2015 Model by fixing the stage-1 and stage-2 selectivity parameters and the catchability coefficient ($q$) for the ADF&G pot survey at those values estimated in the 2015 model (and allows the NMFS trawl survey selectivity to exceed 1 for stage-2 crab). The parameters that are estimated in this model include the average recruitment ($\bar{R}$), the recruitment deviations ($\delta^R_y$), the initial numbers in each stage ($\boldsymbol{n}^0$), the natural mortality deviation 1998 ($\delta^M_{1998}$), and the fishing mortalities for the directed pot fishery, the trawl bycatch fishery, and the fixed bycatch fishery ($\bar{F}^\text{df}$, $\bar{F}^\text{tb}$, $\bar{F}^\text{fb}$, $\delta^\text{df}_{t,y}$, $\delta^\text{tb}_{t,y}$, $\delta^\text{fb}_{t,y}$). As in the 2015 model, the robust multinomial distribution was used to model the length-frequency data.
Gmacs base: directed pot, NMFS trawl survey and ADF&G pot survey selectivities are estimated for stage-1 and stage-2 crab (and fixed at 1 for stage-3 crab). These selectivities are bounded so that they cannot be greater than 1. This model also estimates the catchability coefficient ($q$) for the ADF&G pot survey as well as the average recruitment ($\bar{R}$), the recruitment deviations ($\delta^R_y$), the initial numbers in each stage ($\boldsymbol{n}^0$), the natural mortality deviation 1998 ($\delta^M_{1998}$), and the fishing mortalities for the directed pot fishery, the trawl bycatch fishery, and the fixed bycatch fishery ($\bar{F}^\text{df}$, $\bar{F}^\text{tb}$, $\bar{F}^\text{fb}$, $\delta^\text{df}_{t,y}$, $\delta^\text{tb}_{t,y}$, $\delta^\text{fb}_{t,y}$). As in the 2015 model, the robust multinomial distribution was used to model the length-frequency data.
Gmacs M: is the same as above except that natural mortality ($M$) is fixed at 0.18 $\text{yr}^{-1}$ during all years.
Gmacs Francis: is similar to the scenario above except that it also uses the Francis iterative re-weighting method (Francis 2011), to re-weight the size-composition data relative to the abundance indices. The trawl survey and pot survey weights were left as is (i.e. a weight of 1) because upweighting these series resulted in worse standard deviation of the normalised residual (SDNR) and median of the absolute residual (MAR) values for each of the surveys. Down-weighting the two surveys actually improved the SDNR and MAR values, but it would be unwise to down-weight either of these series. When applying the Francis iterative re-weighting method only once iteration was done (i.e. the model was run once with the size composition likelihood weights set to one, the new Francis weights were calculated, and the model was run once more using these weights). In this scenario the multinomial distribution was used instead as the theory underpinning the Francis weighting method is based on this distribution.
Gmacs force: is an exploratory scenario that the same as above except the NMFS trawl survey is up-weighted by $\lambda^\text{NMFS}=$ 1 and the ADF&G pot survey is up-weighted by $\lambda^\text{ADFG}=$ 1. After this, the Francis weights for each of the size-compostitons were recalculated and applied again in this model. This scenario should not be used for overfishing determination as it upweights the trawl and pot survey abundance indices to force a better fit to each of these data sets and provide some contrast among the Gmacs model runs. This scenario forces a better fit to the trawl and pot surveys at the expense of the SDNR (and MAR) for each of these series.

Table outlines the major features of each of the models.

Results

Results for all Gmacs scenarios are provided with comparisons to the 2015 model. We recommend the Gmacs base scenario for management purposes since it provides the best fit to the data and is most consistent with previous model specifications.

a. Effective sample sizes and weighting factors.

Observed and estimated effective sample sizes are compared in Table . Effective sample sizes are also shown on size-composition plots (Figures , , and ).

Data weighting factors, SDNRs, and MARs are presented in Table . The SDNR for the trawl survey is acceptable at 1.44 in the Gmacs match scenario, and improves to 1.41 in the Gmacs base scenario. In the Gmacs M model the SDNR of the trawl survey is slightly worse at 1.59, and is much worse in the exploratory Gmacs force scenario at 2.16. The SDNRs for the pot surveys show much the same pattern between each of the scenarios, but are much higher values (ranging from 3.95 to 5.19). These values are very high, and whilst they can be improved by down-weighting the pot survey, it is recommended that they be left as they are as the pot survey is one of the most important data series in this model. The MAR for the trawl and pot surveys shows the same pattern among each of the scenarios as the SDNR. The SDNR (and MAR) values for the trawl survey and pot survey size compositions were excellent, ranging from 0.78 to 1.30 (except for in the Gmacs force scenario where the weights were a little high). The SDNRs for the directed pot fishery size compositions are a little low, ranging from 0.64 to 0.79. However, the SDNRs (and MARs) were not used when weighting the size composition data sets in those scenarios that used the Francis weighting method (i.e. in the Gmacs Francis, and Gmacs Force scenarios). Instead, the Francis size composition weights were used (Francis 2011).

b. Tables of estimates.

Model parameter estimates for each of the Gmacs scenarios are summarized in Tables , , , , and . These parameter estimates are compared in Table . Negative log-likelihood values and management measures for each of the Gmacs scenarios are compared in Tables and .

There is little difference in the parameter estimates within the Gmacs match and Gmacs base scenarios. This is reflected in the log-likelihood components and the management quantities. The parameter estimates in the Gmacs M scenario are a little different to the previous scenarios, particularly the estimate of the ADF&G pot survey catchability ($q$) (see Table ).

c. Graphs of estimates.

Estimated (and fixed) selectivities are compared in Figure .

The various model fits to total male ($>$ 89 mm CL) trawl survey biomass are compared in Figures and . The fits to pot survey CPUE are compared in Figures and . Standardized residuals of total male trawl survey biomass and pot survey CPUE are plotted in Figures and .

Fits to stage compositions for trawl survey, pot survey, and commercial observer data are shown in Figures , , and for the all scenarios. Bubble plots of stage composition residuals for trawl survey, pot survey, and commercial observer data are shown for the Gmacs base, Gmacs M, Gmacs Francis, and Gmacs force scenarios in Figures , , , and , respectively.

Fits to retained catch numbers and bycatch biomass are shown for all Gmacs scenarios in Figure .

Estimated recruitment is compared in Figure . Estimated abundances by stage and mature male biomasses for all scenarios (including the 2015 model) are shown in Figures and . Estimated natural mortality each year ($M_t$) is presented in Figure .

d. Graphic evaluation of the fit to the data.

There is little difference between model estimated survey biomass in the gmacs scenarios when compared with the 2015 model (Figures and ). Looking at the model fits to the NMFS trawl survey biomass (Figure ), the Gmacs match scenario is the most similar to the 2015 model, and the Gmacs base model is very similar as well. In all scenarios, Gmacs produces a better fit during the mid-late 1980s. However, since about 2010 Gmacs estimates a slighly lower survey biomass than the 2015 model in an attempt to better fit the ADF&G pot survey CPUE (Figure ). The three Gmacs scenarios that do not attempt to estimate natural mortality in 1998/99 (Gmacs M, Gmacs Francis, and Gmacs force) predict lower survey biomass from 1992 to 1998 than the other scenarios and the 2015 model. These same two runs also predict a lower survey biomass in recent years (since about 2010). While these two models may result in slightly worse fits to the data, they do not risk over-fitting the data in the same way the other scenarios do. As exptected the model that upweights the NMFS survey biomass and ADF&G pot survey CPUE (Gmacs force) provides a better fit to the survey biomass during the mid-late 1980s and a much better fit to the pot survey CPUE in the most recent two years (Figures , , , and ). Keep in mind that this scenario was only included for exploratory purposes and forcing these weights resulted in worse SDNR and MAR values for the two abundance indices.

Estimated recruitment to the model is variable over time (Figure ). Estimated recruitment during recent years is generally low in all scenarios. Estimated mature male biomass on 15 February also fluctuates strongly over time (Figure ).

e. Retrospective and historic analyses.

Gmacs retrospective analyses under development.

f. Uncertainty and sensitivity analyses.

Estimated standard deviations of parameters and selected management measures for the five Gmacs scenarios are summarized in Tables , , , , and . Probabilities for mature male biomass and OFL in 2016 are illustrated in Section F.

g. Comparison of alternative model scenarios.

Both the Gmacs match and Gmacs base scenarios provide adequate matches between the 2015 model and its Gmacs equivalent. In fact, despite a few minor differences, estimates produced by the 2015 model are generally encompassed the in the uncertainty bounds of the Gmacs match model.

Looking at the plot of mature male biomass (Figure ), the Gmacs force scenario stands out as being quite different to the other models (including the 2015 model). This scenario results in a lower MMB from the mid-1908s through to the late-1990s, and is again lower in the most recent 5 years. This scenario upweights both the trawl survey and the pot survey abundance indices (it upweights the pot survey more than the trawl survey) and represents a model run that places greater trust in the abundance indices, particularly the pot survey, than other data sources.

Although the Gmacs M scenario presents a worse fit to the data, particularly the NMFS trawl-survey time series, this model does not simply allow a better fit to by estimating an unconstrained pulse in natural mortality. Although doing so produces a better fit to the model, it reduces predictive power and support for such a phenomena, anecdotal or otherwise, seems to be limited. It also raises concerns about what the implications would be for an “average” true natural mortality which can affect the management measures. Despite these concerns, more work is needed in the future to explore more parsimonious alternatives that provide better fits to the data.

In summary, we recommend the Gmacs base scenario for management purposes since it provides the best fit to the data and is most consistent with previous model specifications. Our initial preference was for Gmacs M since we had difficulty justifying an abrubt, single-year anomaly in natural mortality. However, the fact that the residual pattern is worse and until further work can be completed on alternative model specifications (e.g., better accounting of spatial processes affecting the data), the Gmacs base model was considered reasonable and should be used for overfishing determination for this stock in 2016.

F. Calculation of the OFL and ABC

The overfishing level (OFL) is the fishery-related mortality biomass associated with fishing mortality $F_\mathit{OFL}$. The SMBKC stock is currently managed as Tier 4 (2013 SAFE), and only a Tier 4 analysis is presented here. Thus given stock estimates or suitable proxy values of $B_\mathit{MSY}$ and $F_\mathit{MSY}$, along with two additional parameters $\alpha$ and $\beta$, $F_\mathit{OFL}$ is determined by the control rule \[\begin{align} F_\mathit{OFL} &= \begin{cases} F_\mathit{MSY}, &\text{ when } B/B_\mathit{MSY} > 1\\ F_\mathit{MSY} \frac{\left( B/B_\mathit{MSY} - \alpha \right)}{(1 - \alpha)}, &\text{ when } \beta < B/B_\mathit{MSY} \le 1 \end{cases}\\ F_\mathit{OFL} &< F_\mathit{MSY} \text{ with directed fishery } F = 0 \text{ when } B/B_\mathit{MSY} \le \beta \notag \end{align}\]

where $B$ is quantified as mature-male biomass (MMB) at mating with time of mating assigned a nominal date of 15 February. Note that as $B$ itself is a function of the fishing mortality $F_\mathit{OFL}$ (therefore numerical approximation of $F_\mathit{OFL}$ is required). As implemented for this assessment, all calculations proceed according to the model equations given in Appendix A. $F_\mathit{OFL}$ is taken to be full-selection fishing mortality in the directed pot fishery and groundfish trawl and fixed-gear fishing mortalities set at their model geometric mean values over years for which there are data-based estimates of bycatch-mortality biomass.

The currently recommended Tier 4 convention is to use the full assessment period, currently 1984-2016, to define a $B_\mathit{MSY}$ proxy in terms of average estimated MMB and to set $\gamma$ = 1.0 with assumed stock natural mortality $M$ = 0.18 $\text{yr}^{-1}$ in setting the $F_\mathit{MSY}$ proxy value $\gamma M$. The parameters $\alpha$ and $\beta$ are assigned their default values $\alpha$ = 0.10 and $\beta$ = 0.25. The $F_\mathit{OFL}$, OFL, ABC, and MMB in 2016 for all scenarios are summarized in Table . ABC is 80% of the OFL.

G. Rebuilding Analysis

This stock is not currently subject to a rebuilding plan.

H. Data Gaps and Research Priorities

Growth increments and molting probabilities as a function of size.
Trawl survey catchability and selectivities.
Temporal changes in spatial distributions near the island.
Natural mortality.

I. Projections and Future Outlook

With the decline of estimated population biomass during recent years, outlook for this stock is not promising. If the decline continues, the stock will fall to depleted status soon.

J. Acknowledgements

We thank the Crab Plan Team, Doug Pengilly for reviewing the earlier draft of this manuscript. Some materials in the report are from the SAFE report prepared by Bill Gaeuman in 2014. We thank Andre Punt for his input into the Gmacs model and for finding the error in the old SMBKC model code.

K. References

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Jensen, G.C. and D.A. Armstrong. 1989. Biennial reproductive cycle of blue king crab, Paralithodes platypus, at the Pribilof Islands, Alaska and comparison to a congener, P. camtschatica. Can. J. Fish. Aquat. Sci. 46: 932-940.

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$Comparisons of the estimated (and fixed to match the 2015 model selectivities in the Gmacs base scenario) stage-1 and stage-2 selectivities for each of the different model scenarios (the stage-3 selectivities are all fixed at 1). Estimated selectivities are shown for the directed pot fishery, the trawl bycatch fishery, the fixed bycatch fishery, the NMFS trawl survey, and the ADF&G pot survey. Two selectivity periods are estimated in the directed pot fishery, from 1978-2008 and 2009-2016.\label{fig:molt_prob}$

Comparisons of the estimated (and fixed to match the 2015 model selectivities in the Gmacs base scenario) stage-1 and stage-2 selectivities for each of the different model scenarios (the stage-3 selectivities are all fixed at 1). Estimated selectivities are shown for the directed pot fishery, the trawl bycatch fishery, the fixed bycatch fishery, the NMFS trawl survey, and the ADF&G pot survey. Two selectivity periods are estimated in the directed pot fishery, from 1978-2008 and 2009-2016.

$Probability of growth transition by stage. Each of the panels represent the stage before a transition. The x-axes represent the stage after a transition. The size transition matrix was provided as an input directly to Gmacs (as it was during the 2015 SMBKC assessment).\label{fig:growth_trans}$

Probability of growth transition by stage. Each of the panels represent the stage before a transition. The x-axes represent the stage after a transition. The size transition matrix was provided as an input directly to Gmacs (as it was during the 2015 SMBKC assessment).

$Comparisons of area-swept estimates of total male survey biomass (tons) and model predictions for the 2015 model and each of the Gmacs model scenarios. The error bars are plus and minus 2 standard deviations.\label{fig:trawl_survey_biomass}$

Comparisons of area-swept estimates of total male survey biomass (tons) and model predictions for the 2015 model and each of the Gmacs model scenarios. The error bars are plus and minus 2 standard deviations.

$Comparisons of area-swept estimates of total male survey biomass (tons) and model predictions for the 2015 model and each of the Gmacs model scenarios. The solid black error bars are plus and minus 2 standard deviations derived using the original survey CVs. The dotted error bars are plus and minus 2 standard deviations but represent the weighted survey CVs.\label{fig:trawl_survey_biomass_weights}$

Comparisons of area-swept estimates of total male survey biomass (tons) and model predictions for the 2015 model and each of the Gmacs model scenarios. The solid black error bars are plus and minus 2 standard deviations derived using the original survey CVs. The dotted error bars are plus and minus 2 standard deviations but represent the weighted survey CVs.

$Standardized residuals for area-swept estimates of total male survey biomass for each of the Gmacs model scenarios. \label{fig:bts_resid_nmfs}$

Standardized residuals for area-swept estimates of total male survey biomass for each of the Gmacs model scenarios.

$Comparison of observed and model predicted retained catch and bycatches in each of the Gmacs models. Note that difference in units between each of the panels, some panels are expressed in numbers of crab, some as biomass (tons).\label{fig:fit_to_catch}$

Comparison of observed and model predicted retained catch and bycatches in each of the Gmacs models. Note that difference in units between each of the panels, some panels are expressed in numbers of crab, some as biomass (tons).

$Comparisons of estimated recruitment time series during 1979-2016 in each of the scenarios. The solid horizontal lines in the background represent the estimate of the average recruitment parameter ($\bar{R}$) in each model scenario.\label{fig:recruitment}$

Comparisons of estimated recruitment time series during 1979-2016 in each of the scenarios. The solid horizontal lines in the background represent the estimate of the average recruitment parameter ($\bar{R}$) in each model scenario.

$Comparisons of estimated mature male biomass (MMB) time series on 15 February during 1978-2016 for each of the model scenarios.\label{fig:mmb}$

Comparisons of estimated mature male biomass (MMB) time series on 15 February during 1978-2016 for each of the model scenarios.

$Distribution of carapace width (mm) at recruitment.\label{fig:init_rec}$

Distribution of carapace width (mm) at recruitment.

$Numbers by stage each year (at the beginning of the model year, i.e. 1 July, season 1) in each of the models including the 2015 model.\label{fig:init_N}$

Numbers by stage each year (at the beginning of the model year, i.e. 1 July, season 1) in each of the models including the 2015 model.

$Time-varying natural mortality ($M_t$). Estimated pulse period occurs in 1998/99 (i.e. $M_{1998}$). \label{fig:M_t}$

Time-varying natural mortality ($M_t$). Estimated pulse period occurs in 1998/99 (i.e. $M_{1998}$).

Appendix A: SMBKC Model Description

1. Introduction

The Gmacs model has been specified to account only for male crab at least 90 mm in carapace length (CL). These are partitioned into three stages (size-classes) determined by CL measurements of (1) 90-104 mm, (2) 105-119 mm, and (3) 120+ mm. For management of the St. Matthew Island blue king crab (SMBKC) fishery, 120 mm CL is used as the proxy value for the legal measurement of 5.5 mm in carapace width (CW), whereas 105 mm CL is the management proxy for mature-male size (5 AAC 34.917 (d)). Accordingly, within the model only stage-3 crab are retained in the directed fishery, and stage-2 and stage-3 crab together comprise the collection of mature males. Some justification for the 105 mm value is presented in Pengilly and Schmidt (1995), who used it in developing the current regulatory SMBKC harvest strategy. The term “recruit” here designates recruits to the model, i.e., annual new stage-1 crab, rather than recruits to the fishery. The following description of model structure reflects the Gmacs base model configuration.

2. Model Population Dynamics

Within the model, the beginning of the crab year is assumed contemporaneous with the NMFS trawl survey, nominally assigned a date of 1 July. Although the timing of the fishery is different each year, MMB is measured 15 February, which is the reference date for calculation of federal management biomass quantities. To accommodate this, each model year is split into 4 seasons ($t$) and a proportion of the natural mortality ($\tau_t$) is applied in each of these seasons where $\sum_{t=1}^{t=5} \tau_t = 1$. Each model year consists of the following processes:

The proportion of natural mortality ($\tau_t$) applied during each season in the model is provided in Table . The beginning of the year (1 July) to the date that MMB is measured (15 February) is 63% of the year. Therefore 63% of the natural mortality must be applied before the MMB is calculated. Because the timing of the fishery is different each year $\tau_2$ is different each year and thus $\tau_4$ differs each year.

With boldface lower-case letters indicating vector quantities we designate the vector of stage abundances during season $t$ and year $y$ as \[\begin{equation} \boldsymbol{n}_{t,y} = n_{l,t,y} = \left[ n_{1,t,y}, n_{2,t,y}, n_{3,t,y} \right]^\top. \end{equation}\] The number of new crab, or recruits, of each stage entering the model each season $t$ and year $y$ is represented as the vector $\boldsymbol{r}_{t,y}$. The SMBKC formulation of Gmacs specifies recruitment to stage-1 only during season $t=5$, thus the recruitment size distribution is \[\begin{equation} \phi_l = \left[ 1, 0, 0 \right]^\top, \end{equation}\] and the recruitment is \[\begin{equation} \boldsymbol{r}_{t,y} = \begin{cases} 0 &\text{for} \quad t<5\\ \bar{R} \phi_l \delta^R_y &\text{for} \quad t=5. \end{cases} \end{equation}\] where $\bar{R}$ is the average annual recruitment and $\delta^R_y$ are the recruitment deviations each year $y$ \[\begin{equation} \delta^R_y \sim \mathcal{N} \left( 0, \sigma_R^2 \right). \end{equation}\] Using boldface upper-case letters to indicate a matrix, we describe the size transition matrix $\boldsymbol{G}$ as \[\begin{equation} \boldsymbol{G} = \left[ \begin{array}{ccc} 1 - \pi_{12} - \pi_{13} & \pi_{12} & \pi_{13} \\ 0 & 1 - \pi_{23} & \pi_{23} \\ 0 & 0 & 1 \end{array} \right], \end{equation}\]

with $\pi_{jk}$ equal to the proportion of stage-$j$ crab that molt and grow into stage-$k$ within a season or year.

The natural mortality each season $t$ and year $y$ is \[\begin{equation} M_{t,y} = \bar{M} \tau_t + \delta_y^M \text{ where } \delta_y^M \sim \mathcal{N} \left( 0, \sigma_M^2 \right) \end{equation}\] Fishing mortality by year $y$ and season $t$ is denoted $F_{t,y}$ and calculated as \[\begin{equation} F_{t,y} = F_{t,y}^\text{df} + F_{t,y}^\text{tb} + F_{t,y}^\text{fb} \end{equation}\] where $F_{t,y}^\text{df}$ is the fishing mortality associated with the directed fishery, $F_{t,y}^\text{tb}$ is the fishing mortality associated with the trawl bycatch fishery, $F_{t,y}^\text{fb}$ is the fishing mortality associated with the fixed bycatch fishery. Each of these are derived as \[\begin{align} F_{t,y}^\text{df} &= \bar{F}^\text{df} + \delta^\text{df}_{t,y} \quad \text{where} \quad \delta^\text{df}_{t,y} \sim \mathcal{N} \left( 0, \sigma^2_\text{df} \right), \notag\\ F_{t,y}^\text{tb} &= \bar{F}^\text{tb} + \delta^\text{tb}_{t,y} \quad \text{where} \quad \delta^\text{df}_{t,y} \sim \mathcal{N} \left( 0, \sigma^2_\text{tb} \right), \notag\\ F_{t,y}^\text{fb} &= \bar{F}^\text{fb} + \delta^\text{fb}_{t,y} \quad \text{where} \quad \delta^\text{df}_{t,y} \sim \mathcal{N} \left( 0, \sigma^2_\text{fb} \right), \end{align}\] where $\delta^\text{df}_{t,y}$, $\delta^\text{tb}_{t,y}$, and $\delta^\text{fb}_{t,y}$ are the fishing mortality deviations for each of the fisheries, each season $t$ during each year $y$, $\bar{F}^\text{df}$, $\bar{F}^\text{tb}$, and $\bar{F}^\text{fb}$ are the average fishing mortalities for each fishery. The total mortality $Z_{l,t,y}$ represents the combination of natural mortality $M_{t,y}$ and fishing mortality $F_{t,y}$ during season $t$ and year $y$ \[\begin{equation} \boldsymbol{Z}_{t,y} = Z_{l,t,y} = M_{t,y} + F_{t,y}. \end{equation}\] The survival matrix $\boldsymbol{S}_{t,y}$ during season $t$ and year $y$ is \[\begin{equation} \boldsymbol{S}_{t,y} = \left[ \begin{array}{ccc} 1-e^{-Z_{1,t,y}} & 0 & 0 \\ 0 & 1-e^{-Z_{2,t,y}} & 0 \\ 0 & 0 & 1-e^{-Z_{3,t,y}} \end{array} \right]. \end{equation}\] The basic population dynamics underlying Gmacs can thus be described as \[\begin{align} \boldsymbol{n}_{t+1,y} &= \boldsymbol{S}_{t,y} \boldsymbol{n}_{t,y}, &\text{ if } t<5 \notag\\ \boldsymbol{n}_{t,y+1} &= \boldsymbol{G} \boldsymbol{S}_{t,y} \boldsymbol{n}_{t,y} + \boldsymbol{r}_{t,y} &\text{ if } t=5. \end{align}\]

3. Model Data

Data inputs used in model estimation are listed in Table .

4. Model Parameters

Table lists fixed (externally determined) parameters used in model computations. In all scenarios, the stage-transition matrix is \[\begin{equation} \label{eq:size_transition} \boldsymbol{G} = \left[ \begin{array}{ccc} 0.2 & 0.7 & 0.1 \\ 0 & 0.4 & 0.6 \\ 0 & 0 & 1 \end{array} \right] \end{equation}\] which is the combination of the growth matrix and molting probabilities. Estimated parameters are listed in Table and include an estimated natural mortality deviation parameter in 1998/99 ($\delta^M_{1998}$) assuming an anomalous mortality event in that year, as hypothesized by Zheng and Kruse (2002), with natural mortality otherwise fixed at 0.18 $\text{yr}^{-1}$.

5. Model Objective Function and Weighting Scheme

The objective function consists of the sum of several “negative log-likelihood” terms characterizing the hypothesized error structure of the principal data inputs (Table ). A lognormal distribution is assumed to characterize the catch data and is modelled as \[\begin{align} \sigma_{t,y}^\text{catch} &= \sqrt{\log \left( 1 + \left( \mathit{CV}_{t,y}^\text{catch} \right)^2 \right)}\\ \delta_{t,y}^\text{catch} &= \mathcal{N} \left( 0, \left( \sigma_{t,y}^\text{catch} \right)^2 \right) \end{align}\] where $\delta_{t,y}^\text{catch}$ is the residual catch. The relative abundance data is also assumed to be lognormally distributed \[\begin{align} \sigma_{t,y}^\text{I} &= \frac{1}{\lambda} \sqrt{\log \left( 1 + \left( \mathit{CV}_{t,y}^\text{I} \right)^2 \right)}\\ \delta_{t,y}^\text{I} &= \log \left( I^\text{obs} / I^\text{pred} \right) / \sigma_{t,y}^\text{I} + 0.5 \sigma_{t,y}^\text{I} \end{align}\] and the likelihood is \[\begin{equation} \sum \log \left( \delta_{t,y}^\text{I} \right) + \sum 0.5 \left( \sigma_{t,y}^\text{I} \right)^2 \end{equation}\]

Gmacs calculates standard deviation of the normalised residual (SDNR) values and median of the absolute residual (MAR) values for all abundance indices and size compositions to help the user come up with resonable likelihood weights. For an abundance data set to be well fitted, the SDNR should not be much greater than 1 (a value much less than 1, which means that the data set is fitted better than was expected, is not a cause for concern). What is meant by “much greater than 1” depends on $m$ (the number of years in the data set). Francis (2011) suggests upper limits of 1.54, 1.37, and 1.26 for $m$ = 5, 10, and 20, respectively. Although an SDNR not much greater than 1 is a necessary condition for a good fit, it is not sufficient. It is important to plot the observed and expected abundances to ensure that the fit is good.

Gmacs also calculates Francis weights for each of the size composition data sets supplied (Francis 2011). If the user wishes to use the Francis iterative re-weighting method, first the weights applied to the abundance indices should be adjusted by trial and error until the SDNR (and/or MAR) are adequte. Then the Francis weights supplied by Gmacs should be used as the new likelihood weights for each of the size composition data sets the next time the model is run. The user can then iteratively adjust the abundance index and size composition weights until adequate SDNR (and/or MAR) values are achieved, given the Francis weights.

6. Estimation

The model was implemented using the software AD Model Builder (Fournier et al. 2012), with parameter estimation by minimization of the model objective function using automatic differentiation. Parameter estimates and standard deviations provided in this document are AD Model Builder reported values assuming maximum likelihood theory asymptotics.

A correction to the 2015 model code was made in the population dynamics function involving how the growth transition matrix was applied to the numbers at length to calculate the numbers during the following time-step, specifically `N(t+1,3)=TM(2,3)*NN(2)+NN(3);` was changed to `N(t+1,3)=TM(1,3)*NN(1)+TM(2,3)*NN(2)+NN(3);`.↩